Here's what I would do: your equations are actually linear in the variables 1/a, 1/b, 1/c, and 1/d. So define new variables equal to those reciprocals. Then use standard methods to solve for the reciprocal variables. Finally, flip them over again to get your a, b, c, and d.
Question: are the coefficients of the matrix going to stay the same? Or do all the coefficients change for your repeat problems?